What is meant by the expected BLEU cost when training with BLEU and SIMILE?
It looks like the method they use for training takes a set of candidate hypotheses $\mathcal{U}(x)$, along with associated probabilities, and then minimizes the expected value of the cost function over that distribution. Section 3 has the loss function being minimized: 

$ \mathcal{L}_{Risk} = \sum\limits_{u \in \mathcal{U}(x)} cost(t, u) \frac{p(u|x)}{\sum_{u' \in \mathcal{U}(x)} p(u'|x)} $. 

One of the cost functions used is $1 - \texttt{BLEU}(t, h)$, where $t$ is the target and $h$ is the generated hypothesis. I'm not sure where $p(u|x)$ is coming from, but $1 - \mathcal{L}_{Risk}$ for the BLEU cost function is probably what they're refering to when they mention Expected BLEU.